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A friend of mine just showed me this article about the "how old is the shepherd" problem: Link

Of course, I'm shocked by the finding that 75 percent of kids give an answer other than "there isn't enough information." But I wonder whether the fustiness of the example may have had something to do with it: shepherds and flocks of sheep and dogs aren't exactly commonplace for most American schoolchildren.

My question: Does anyone know of a similar study that uses terms and concepts that might be more familiar to schoolchildren. Something like "There are 14 girls and 9 boys in a class. How old is the teacher?" (Maybe that one is even a little behind the times, too.)

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    $\begingroup$ Some additional info: the original 1986 study was done with first and second grade Swiss children, not Americans. I think Swiss children would be much more familiar with sheep. $\endgroup$ – Scott Eberle Oct 18 '16 at 10:36
  • $\begingroup$ @ScottEberle Do you have a citation? I had believed it was from 1989 (see my post below) but would happily correct it to '86 if that is the case. $\endgroup$ – Benjamin Dickman Oct 18 '16 at 14:33
  • $\begingroup$ The link is in the second paragraph of the article in the question. The linked paper says it was to be presented at the AERA meeting in April 1986. $\endgroup$ – Scott Eberle Nov 9 '16 at 17:50
  • $\begingroup$ I beg to differ[...] I think the answer of 36 was extremely intelligent, far more so than the questioners. For it was clearly a BS question, and how do you answer a BS question? With “I don’t know”? Of course not, that’s a losing move. With “that question’s BS, shame on you”? Certainly not! No, the most practical way to answer a BS question is with more BS. You might want to tell the boss what he needs to hear, but you need to tell the boss what he wants to hear. They tut-tut at their students’ lack of critical thought; but it was they who were conspicuously clueless. $\endgroup$ – Nathaniel Hellerstein Feb 13 '18 at 1:48
  • $\begingroup$ I'm actually somewhat sympathetic to this. I wonder if the kids would react the same way if the question came up more spontaneously rather than being asked by an authority figure. Maybe it says more about how we train kids to think of teachers and other authority figures as infallible. $\endgroup$ – Tim kinsella Feb 13 '18 at 20:45
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Note (Feb 2018): There is an alleged "Chinese math problem" (see, e.g., WaPo article) going around about the second example problem below (cited to Reusser 1988, but can be found in Reusser 1986, as I've tweeted here). Interesting that it has gone viral without anyone having sourced it.


This study can be easily replicated, and has been: with multiple scenarios, and in multiple languages. One keyword to use in the context of the mathematics education literature is sense making, and one author to whom you might look is Alan Schoenfeld; for example, see "A Modest Proposal" from the Notices of the AMS.

The linked paper provides several examples, including the shepherd one (I believe the original citation for that phrasing is Barsuk (1989); or, at least, that is what I asserted in a comment two years ago). You may as well click through to read the write-up, but here is an excerpt to keep this self-contained:

enter image description here

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    $\begingroup$ But/and I myself still do not know why kids (and adults) will both believe that authority/math is nonsensical and try to comply, rather than "raise critical objections". Presumably a game-theoretic sort of optimization? $\endgroup$ – paul garrett Oct 17 '16 at 23:17
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    $\begingroup$ @paulgarrett My guess is that students (of all ages) tire of "mathematics" woven into contrived formulations and, to unwind the arithmetic underpinnings rapidly, they are content to pick out numbers and guess an operation (similar to a game). Hence Schoenfeld's call for sense-making. Some of this will also be related to how/when/whether we reward sense-making; I am reminded of a tale Henry Pollak enjoys telling in person: link from p. 40 of here. $\endgroup$ – Benjamin Dickman Oct 17 '16 at 23:32
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    $\begingroup$ I feel like this last example (a statement with no question) is something I routinely see in other teacher's tests and practice materials. The majority of my community-college students will refer to an equation as a "question". Flipping to our custom arithmetic text, 12 of 46 problems on the final practice (as an example) are naked expressions with no text direction or question. It's frustratingly endemic. $\endgroup$ – Daniel R. Collins Oct 18 '16 at 0:20
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    $\begingroup$ @BenjaminDickman Another possible explanation: generally, we punish students whenever they display a lack of understanding of something. This leads to students trying to hide their ignorance. Generally teachers ask "reasonable" questions, so when an unreasonable one sneaks in they just assume that they are "too dumb" to get it, and they fake something. $\endgroup$ – Steven Gubkin Oct 20 '16 at 3:22
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    $\begingroup$ @mweiss As mentioned in my tweet [ twitter.com/benjamindickman/status/961629845007552518 ] I believe that the "original" sourcing goes back at least to the 1840s [although it was not explicitly in the context of Mathematics Education]. See, e.g.: fr.wiktionary.org/wiki/%C3%A2ge_du_capitaine $\endgroup$ – Benjamin Dickman Feb 19 '18 at 3:23
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Briefly, I'd say, "yes" to your question. Even for more relatable questions, the students still draw the same sort of invalid answer.

I find it a stretch that students can't understand the shepherd problem. If they can't relate to it, then we've done a very poor job of educating them about the world, history, and a lot of other things which have very little to do with mathematics.

Nevertheless, I don't fully agree with the analysis of the linked article either. Ask the students a completely different question, "There are 125 sheep and 5 dogs in a flock. What is the shepherd's name?" Whether or not 75% will give you an answer, you will still see a disappointingly large percentage of students providing you an answer.

Student, "The shepherd's name is John."

Teacher, "How did you come to that conclusion?"

Student, "Well, it could be."

Let's just ignore that the student didn't answer the teacher's question. I've encountered this scenario numerous times in the classroom where a question doesn't really have an answer, so the students make something up, and, because the answer has some kind of plausibility to it, the students think they're done.

The way I would work with these sorts of problems is to pepper them throughout the course. But also, I would focus on the reasoning skills the students need to solve the problem. So, you can have a discussion of what's sorts of information you need to determine someone's age. This can lead to all sorts of interesting answers. For example, many students will offer, "their birthday." The students can relatively quickly figure out that knowing a birthday is insufficient; eventually they'll come up with, "their birth date" or something equivalent. And then the teacher can ask, "do you need to know their birth date to determine their age?" This becomes more challenging for them because the previous answer has locked many of their minds into one way of thinking about the problem. Yet, with some patience and good give-and-take between the teacher and the students, they should be able to see that you can answer such questions as, "My sister was born when my mom was 30 years old. My mom is now 45. How old is my sister?"

I am personally fond of questions like "Jackie's brother is five years older than Paul. Paul was born in 2002. How old is Jackie's brother?" This isn't quite like the shepherd problem, but it's similar in that there isn't quite enough information to know with 100% certainty. However, there are two plausible answers.

So back to the teacher's question, "how did you come to that conclusion?"

The students need to be taught how to explain how they're getting their answers. In an English class, saying something like, "It could be John", might be perfectly acceptable (I'm not an English teacher so I won't weigh in too strongly about the validity of that answer). But, in mathematics and often in the sciences, a plausible answer is usually not what we're looking for. We're usually looking for reasoning that starts for a collection of facts and draws various conclusions based upon those facts.

So, it might be plausible that the shepherd is 25 years old. It might just happen that way. Great. But is this supported by the facts? I would approach this as outlined above: by stepping back and having the students review what sort of reasoning skills they need to use and how fact relate to the sorts of answers they're trying to come up with. And then once you've perhaps convinced them that we can't answer a question about the shepherd's age, ask the students, "what sorts of questions could we ask about this shepherd from the facts presented?"

Ultimately, the students should be able to answer a question like, "There are 125 gabbalahs and 5 tumps. What is Margaret's age?" If the students give up and say, "I don't know. What's a gabbalah? I don't know what a tump is", then the students are again missing the point of the question. Giving up isn't the same thing as saying the question as presented isn't answerable. In fact, what I'd most want to hear from the students is, "Do gabbalah and tumps have anything to do with figuring out someone's age?" That's a great question.

The younger the student, generally, the more a concrete thinker the student is. It's very challenging for them to think about something abstractly (such as in the gabbalah/tump question). But also, I would say that it's not a failure so much of mathematics education that makes the students think they need to come up with an answer. It's about how we teach the students over all. Our whole system of education is biased toward asking questions that the students can answer. (It's not too hard to see why we do that.) And, this bias in the pedagogy biases the students in how they approach problems.

While I may disagree that the shepherd question is fusty, even if it is, that should not be preventing the students from answering the question. In our pedagogical aim to make problem relevant to students, we lose sight that we should also be expanding their horizons, broadening their perspectives, and showing and teaching them ways of grappling with the unfamiliar. (Too often we're encouraging them to stay within their myopic world views by only presenting what is already known and familiar.) Speaking as a mathematician or a scientist, there is very little of interest done in mathematics (or the sciences) which is commonplace; what is interesting in mathematics and the sciences are those places where we lack knowledge and understanding---where our comfort zone is being challenged.

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  • $\begingroup$ Great answer (+1) I may have some questions for you when I get a minute. Thanks! $\endgroup$ – Tim kinsella Oct 17 '16 at 23:14
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    $\begingroup$ The students appear to be Bayesians! Take all given information into account and reply with the most probable answer given this information. $\endgroup$ – Dirk Oct 18 '16 at 2:18
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Just to add an historical note to the discussion: the original problem was phrased by Flaubert in a letter to his sister and was asking "the age of the captain": Age of the captain.

Nowadays both in Italy and France such kind of nonsense math problem are named as "age of the captain problem". Stella Baruk, a French researcher, published a whole book with this title: "L'âge du capitaine". De l'erreur en mathématiques. Paris, Seuil, 1985. I haven't checked whether it was translated into English.There is also a math book by Hans Freudenthal: Revisiting mathematics education, Kluwer (1991), in which this kind of educational issues in math are adressed.

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